# IJPAM: Volume 61, No. 2 (2010)

**LINEAR CODES AS BINOMIAL IDEALS**

Institute of Computer Technology (E-13)

Hamburg University of Technology

Schwarzenbergstr. 95 E, Hamburg, 21071, GERMANY

e-mail: chughtai@tuhh.de

e-mail: k.zimmermann@tuhh.de

**Abstract.**Recently, binary linear codes were associated with binomial ideals. We show that each linear code can be described by a binomial ideal given as the sum of a toric ideal and a non-prime ideal. We compute the Hilbert polynomials of the projective subschemes corresponding to the binomial ideal of a code and its toric subideal. Moreover, we study the minimal generators and Groebner bases of the binomial ideals of a linear code. The situation turns out to be quite similar to the case of toric ideals. For the binomial ideals of binary linear codes, the Graver bases, the universal Groebner bases, and the set of circuits are essentially equal.

**Received: **February 22, 2010

**AMS Subject Classification: **13P10, 94B05

**Key Words and Phrases: **commutative polynomial rings, binomial ideals, projective subschemes, Groebner bases, Graver bases

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2010

**Volume:** 61

**Issue:** 2